Problem 86 - Calculus for Biology and Medicine

Question

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=2\), assuming that \(f(2)=\) \(-1\) and \(f^{\prime}(2)=1\). $$ y=[f(x)]^{2}-\frac{x}{f(x)} $$

Step-by-Step Solution

Answer

The derivative of \(y\) at \(x=2\) is 1.

Step 1: Applying the Chain Rule

Start by finding the derivative of each part of the function separately. For the first term, \[ y_1 = [f(x)]^2 \] use the chain rule. The derivative is: \[ y_1' = 2f(x) imes f'(x) \] because the derivative of \(u^2\) with respect to \(u\) is \(2u\), and \(u=f(x)\).

Step 2: Applying the Quotient Rule

For the second term, \[ y_2 = \frac{x}{f(x)} \] use the quotient rule, where \(u = x\) and \(v = f(x)\). The derivative is given by: \[ y_2' = \frac{v \cdot u' - u \cdot v'}{v^2} \] Substituting in the values gives \[ y_2' = \frac{f(x) \cdot 1 - x \cdot f'(x)}{[f(x)]^2} = \frac{f(x) - x \cdot f'(x)}{[f(x)]^2} \] since \(u' = 1\) and \(v' = f'(x)\).

Step 3: Combining the Derivatives

Combine the derivatives from Steps 1 and 2 to find \(y'\): \[ y' = y_1' - y_2' = 2f(x)f'(x) - \frac{f(x) - x \cdot f'(x)}{[f(x)]^2} \] This expression represents the derivative of the entire function.

Step 4: Evaluating at \(x=2\)

Substitute \(x=2\), \(f(2)=-1\), and \(f'(2)=1\) into the combined derivative expression: \[ y' = 2(-1)(1) - \frac{-1 - 2(1)}{(-1)^2} \] Simplify: \[ y' = -2 - \frac{-1 - 2}{1} = -2 - (-3) = -2 + 3 = 1 \] Therefore, the derivative at \(x=2\) is \(y' = 1\).

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When you have a function nested inside another function, like in the case of \(y_1 = [f(x)]^2\), the chain rule becomes your go-to method.
A composite function means that you can think of the function as having layers.
In the case of \(y_1 = [f(x)]^2\), the outer layer involves squaring a term, while the inner layer involves the function \(f(x)\).

• The derivative of the outer function, which is \(u^2\) where \(u=f(x)\), is \(2u\).
• Then, we multiply this by the derivative of the inner function \(f(x)\), which is simply \(f'(x)\).

This yields the derivative \(y_1' = 2f(x) \cdot f'(x)\).
Employing the chain rule allows us to deftly tackle intricate functions, offering an elegant solution when direct application of basic differentiation rules would be cumbersome.

Mastering the Quotient Rule

While the chain rule helps with composite functions, the quotient rule is your key to handling fractions, specifically functions in the form of one function divided by another, such as \(y_2 = \frac{x}{f(x)}\).
The quotient rule formula is:

• Let \(u = x\) and \(v = f(x)\).
• Then, the quotient rule formula is: \(\frac{v \cdot u' - u \cdot v'}{v^2}\)
  • Here, \(u'\) is the derivative of \(x\), which is \(1\).
  • And \(v'\) is the derivative of \(f(x)\), which is \(f'(x)\).

This leads to the expression \(y_2' = \frac{f(x) \cdot 1 - x \cdot f'(x)}{[f(x)]^2}\).
This method provides a structured way to differentiate ratios, where handling each part alone could quickly become overly complex. Remembering and applying the quotient rule effectively makes evaluating derivatives of more complex rational functions straightforward and manageable.

The Basics of Differentiation

Differentiation is the process that underlies the aforementioned rules, allowing us to find a rate of change at any given point along a function. It gives us the derivative, which represents the slope of the tangent line to the curve of a function at any point.

• Consider different basic rules of differentiation:
• The power rule: \(\frac{d}{dx}[x^n] = nx^{n-1}\)
• The constant rule: \(\frac{d}{dx}[c] = 0\)

These are some fundamental rules that form the bedrock of more complex derivative calculations, like those using the chain or quotient rule.
In application, differentiation assists in fine-tuning approximations, predicting trends, and managing optimizations across diverse fields.
Whether studying velocity in physics or optimization in economics, differentiation serves as an indispensable tool, illuminating the mechanisms governing dynamic systems.